I can’t guess where this year’s A-to-Z series will lead. Often a theme develops. Complex numbers look like they’re trying to be it. So let me share something from last year’s A-to-Z, and which relies on complex numbers. Julia sets, which are some of the best-known fractals, are calculated by working out functions on complex numbers. By iteration, particularly. That is, start with some number. Evaluate a function where the independent variable has that number. This gets you some (probably) different number. Evaluate the same function again, but using this as the independent variable. This gets you (usually) another number. Evaluate the same function again, with this third number as the independent variable’s value.
You’ve done this sort of iteration when playing with a calculator and hitting the square root or the square or the sine or whatever other function key over and over. These usually end up pretty boring, at 0 or 1 or the calculator reading INF. Put in a slightly different function? You get something beautiful.